Former White House press secretary Dana Perino says the Obama administration should quash any reports that the president's new chief of staff is looking specifically for a woman to replace departing press secretary Robert Gibbs, or else it could risk hurting the next spokesperson's credibility before he or she even walks into the West Wing.

Reports Friday said William Daley, Obama's incoming chief of staff, allegedly wants the next person behind the briefing room podium to be a woman. Former Democratic National Committee spokeswoman Karen Finney is said to be on the short list of possible replacements, as well as current presidential aide Stephanie Cutter.

"The White House shouldn't let that sit out there," Perino, who is also a Fox News contributor, says, "because if they do choose a woman, they don't want to diminish her accomplishment right out of the gate with some who'll think she was chosen just because of her gender."

Perino says whoever the White House hires will have an opportunity to shape the job in his or her own way.

"Each press secretary brings to the job his or her own set of skills and ideas for how best to serve the president, the country, and the press," Perino says. "It's a wonderfully grueling job."